- The system may contain strongly relativistic objects, such as neutron stars or black holes, near and inside which , and the post-Newtonian approximation breaks down. Nevertheless, under some circumstances, the orbital motion may be such that the interbody potential and orbital velocities still satisfy so that a kind of post-Newtonian approximation for the orbital motion might work; however, the strong-field internal gravity of the bodies could (especially in alternative theories of gravity) leave imprints on the orbital motion.
- The evolution of the system may be affected by the emission of gravitational radiation. The 1PN approximation does not contain the effects of gravitational radiation back-reaction. In the expression for the metric given in Box 2, radiation back-reaction effects do not occur until in , in , and in . Consequently, in order to describe such systems, one must carry out a solution of the equations substantially beyond 1PN order, sufficient to incorporate the leading radiation damping terms at 2.5PN order.
- The system may be highly relativistic in its orbital motion, so that even for the interbody field and orbital velocity. Systems like this include the late stage of the inspiral of binary systems of neutron stars or black holes, driven by gravitational radiation damping, prior to a merger and collapse to a final stationary state. Binary inspiral is one of the leading candidate sources for detection by a world-wide network of laser interferometric gravitational wave observatories nearing completion. A proper description of such systems requires not only equations for the motion of the binary carried to extraordinarily high PN orders (at least 3.5PN), but also requires equations for the far-zone gravitational waveform measured at the detector, that are equally accurate to high PN orders beyond the leading ``quadrupole'' approximation.

Of course, some systems cannot be properly described by any post-Newtonian approximation because their behavior is fundamentally controlled by strong gravity. These include the imploding cores of supernovae, the final merger of two compact objects, the quasinormal-mode vibrations of neutron stars and black holes, the structure of rapidly rotating neutron stars, and so on. Phenomena such as these must be analysed using different techniques. Chief among these is the full solution of Einstein's equations via numerical methods. This field of ``numerical relativity'' is a rapidly growing and maturing branch of gravitational physics, whose description is beyond the scope of this article. Another is black hole perturbation theory (see [93] for a review).

Damour [39] calls this the ``effacement'' of the bodies' internal structure. It is a consequence of the strong equivalence principle (SEP), described in Section 3.1.2 .

General relativity satisfies SEP because it contains one and
only one gravitational field, the spacetime metric
. Consider the motion of a body in a binary system, whose size is
small compared to the binary separation. Surround the body by a
region that is large compared to the size of the body, yet small
compared to the separation. Because of the general covariance of
the theory, one can choose a freely-falling coordinate system
which comoves with the body, whose spacetime metric takes the
Minkowski form at its outer boundary (ignoring tidal effects
generated by the companion). There is thus no evidence of the
presence of the companion body, and the structure of the chosen
body can be obtained using the field equations of GR in this
coordinate system. Far from the chosen body, the metric is
characterized by the mass and angular momentum (assuming that one
ignores quadrupole and higher multipole moments of the body) as
measured far from the body using orbiting test particles and
gyroscopes. These asymptotically measured quantities are
oblivious to the body's internal structure. A black hole of mass
*m*
and a planet of mass
*m*
would produce identical spacetimes in this outer region.

The geometry of this region surrounding the one body must be matched to the geometry provided by the companion body. Einstein's equations provide consistency conditions for this matching that yield constraints on the motion of the bodies. These are the equations of motion. As a result the motion of two planets of mass and angular momentum , , and is identical to that of two black holes of the same mass and angular momentum (again, ignoring tidal effects).

This effacement does not occur in an alternative gravitional
theory like scalar-tensor gravity. There, in addition to the
spacetime metric, a scalar field
is generated by the masses of the bodies, and controls the local
value of the gravitational coupling constant (i.e.
*G*
is a function of
). Now, in the local frame surrounding one of the bodies in our
binary system, while the metric can still be made Minkowskian far
away, the scalar field will take on a value
determined by the companion body. This can affect the value of
*G*
inside the chosen body, alter its internal structure
(specifically its gravitational binding energy) and hence alter
its mass. Effectively, each mass becomes several functions
of the value of the scalar field at its location, and several
distinct masses come into play, inertial mass, gravitational
mass, ``radiation'' mass, etc. The precise nature of the
functions will depend on the body, specifically on its
gravitational binding energy, and as a result, the motion and
gravitational radiation may depend on the internal structure of
each body. For compact bodies such as neutron stars, and black
holes these internal structure effects could be large; for
example, the gravitational binding energy of a neutron star can
be 40 percent of its total mass. At 1PN order, the leading
manifestation of this effect is the Nordtvedt effect.

This is how the study of orbiting systems containing compact objects provides strong-field tests of general relativity. Even though the strong-field nature of the bodies is effaced in GR, it is not in other theories, thus any result in agreement with the predictions of GR constitutes a kind of ``null'' test of strong-field gravity.

The Confrontation between General Relativity and
Experiment
Clifford M. Will
http://www.livingreviews.org/lrr-2001-4
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